Droplets, capillary interactions and self-assembly from the equilibrium shape of fluid-fluid interfaces

In this thesis, first we provide a theoretical introduction about fluid-fluid interfaces, and the mathematical models to describe them using a macroscopic approach. Then, we introduce a new numerical method for calculating the equilibrium shape of fluid-fluid interfaces, proving its correctness and pointing out its applicability to study systems of colloidal particles adsorbed at fluid-fluid interfaces, and droplets in contact with solid surfaces, possibly curved and with heterogeneous chemical properties. A very important result presented in this thesis, and obtained through such a new numerical method, is the prediction that capillary interactions can drive cubic particles adsorbed at fluid-fluid interfaces to self-assembly into thermodynamically-stable honeycomb and hexagonal lattices. The capability of experimentally producing honeycomb (i.e. graphene-like) lattices of nanoparticles would be extremely important, and indeed it is currently a very hot research topic, because of the semiconductor properties that these materials would have. Other relevant results presented in this thesis, and obtained from our new numerical method, regard the equilibrium shape of droplets in contact with complex substrates. In particular, we study the equilibrium position of a droplet on a bullet-shaped particle, predicting that the droplet position can shift from the long side of the particle to its flat end just by slightly tuning the bullet geometry. The ability of tuning the droplet position on the particle surface is an important result for the synthesis of odd-shaped colloidal particles, which is a very active field of research. Then, we also study the equilibrium shape of a droplet wetting a flat solid substrate with an ellipsoidal patch with higher wettability than the remaining substrate, showing the different behavior of the droplet with respect to the aspect ratio of the patch shape.